Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2215 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13235 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2)(3,17,22,52)(4,25)(5,32)(6,36)(7,39)(8,42)(9,16)(10,20,45,33)(11,28)(12,18)(13,26)(14,37)(15,40)(19,63,67,114)(21,74,73,47)(23,77,71,111)(24,78,76,44)(27,50)(29,91)(30,94)(31,83)(34,54)(35,97)(38,96)(41,53)(43,100,101,68)(46,104,103,64)(48,92)(49,107)(51,90)(55,85)(56,62)(57,108,105,66)(58,60,98,72)(59,70,95,75)(61,80)(65,118,86,79)(69,109,122,116)(81,84)(82,89)(87,124)(88,99)(93,123)(102,121,115,119)(106,125)(110,112,126,120)(113,127,128,117);(1,3,18,60,41,71,6,21,40,76,85,104,37,75,5,33,9,22,62,72,7,23,38,73,8,24,26,64,34,70,35,20)(2,10,32,95,54,103,55,78,42,74,36,77,39,98,12,52,16,45,97,59,14,46,13,44,15,47,96,111,53,58,56,17)(4,19,61,112,93,100,28,66,92,122,125,127,89,121,27,86,31,67,107,120,29,68,90,108,30,69,84,117,87,119,88,65)(11,43,91,126,80,114,83,118,99,115,82,113,81,116,48,105,51,101,123,110,49,63,25,79,50,102,124,128,106,109,94,57);(1,4,26,84,7,29,5,27,40,92,18,61,34,87,38,90,9,31,85,125,41,93,35,88,8,30,62,107,37,89,6,28)(2,11,36,82,14,49,56,94,42,99,97,123,53,106,55,83,16,51,96,124,54,80,12,48,15,50,32,91,39,81,13,25)(3,19,64,117,23,68,33,86,76,122,60,112,70,119,73,108,22,67,104,127,71,100,20,65,24,69,72,120,75,121,21,66)(10,43,77,113,46,63,17,57,74,115,59,110,58,109,78,118,45,101,111,128,103,114,52,105,47,102,95,126,98,116,44,79)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eg,d^2=ghi,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,h^a=hi,d^b=di,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320566, #72320842, #72320868, … |
| maximal normal subgroup of | #72320566, #72320842, #72320868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60173);